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Suppose that there are multiple martingale measures $Q_1$ and $Q_2$ that attain the minimal variance. Then the convex combination $Q_* := \frac{1}{2}Q_1 + \frac{1}{2}Q_2$ is also a martingale measure. Due to the strict convexity of $f(x) = x^2$, it can be shown that  E_P \left[\frac{dQ_*}{dP}^2 \right] < \frac{1}{2} \left[ \frac{dQ_1}{dP}^2 \right] + ...